Police and judicial sources told AFP they suspect the incident was a deliberate act.
Police have closed two of the metro stations on the Champs-Elysees, a major tourist draw in the French capital.
The incident comes just two months after a policeman was shot and killed on the world-renowned avenue, three days before the first round of France's presidential election.
A note praising the Islamic State group was found next to the body of the gunman, Karim Cheurfi, in that incident.
Police later found other weapons in Cheurfi's car including a shotgun and knives.
France has been hit by a wave of deadly terror attacks in recent years.
The country is still under a state of emergency imposed after the November 2015 attacks in Paris by the Islamic State, when IS jihadists killed 130 people in a night of carnage at venues across the city.
In January 2015, attackers targeted the Charlie Hebdo satirical magazine. In November that year, gunmen and suicide bombers attacked venues around Paris including the Bataclan concert hall, killing 130 people.
Police pulled the driver out of the flaming vehicle and he was 'very likely dead', an interior ministry spokesman said ||Police and judicial sources told AFP?they suspect the incident was a deliberate act